Molecular Bonding. Molecular Schrödinger equation. r - nuclei s - electrons. M j = mass of j th nucleus m 0 = mass of electron
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1 Molecular onding Molecular Schrödinger equation r - nuclei s - electrons 1 1 W V r s j i j1 M j m i1 M j = mass of j th nucleus m = mass of electron j i Laplace operator for nuclei Laplace operator for electrons V e ZZe j j Ze j 4 r 4 r 4 r Coulomb potential ii, ii j, j jj i, j ij electron-electron nuclear-nuclear electron-nuclear repulsion repulsion attraction Copyright Michael D. Fayer, 17
2 orn-oppenheimer pproximation lectrons very light relative to nuclei they move very fast. In the time it takes nuclei to change position a significant amount, electrons have traveled their full paths. Therefore, Fix nuclei - calculate electronic eigenfunctions and energy for fixed nuclear positions. Then move nuclei, and do it again. The resulting curve is the energy as a function of internuclear separation. If there is a minimum bond formation. Copyright Michael D. Fayer, 17
3 orn-oppenheimer pproximation Separation of total Schrödinger equation into an electronic equation and a nuclear equation is obtained by expanding the total Schrödinger equation in powers of M average nuclear mass, m - electron mass. m / M 1/4 1) Not exact ) Good approximation for many problems 3) Many important effects are due to the break down of the orn-oppenheimer approximation. Copyright Michael D. Fayer, 17
4 orn-oppenheimer pproximate wavefunction x, x, n n n electronic coordinates nuclear coordinates nuclear wavefunction depends on electronic quantum number, n and nuclear quantum number, electronic wavefunction depends on electronic quantum number, n Copyright Michael D. Fayer, 17
5 lectronic wavefunction ( x, ) depends on fixed nuclear coordinates,. n Obtained by solving electronic Schrödinger equation for fixed nuclear positions,. No nuclear kinetic energy term. S i M in( x, ) Un( ) V( x, ) n( x, ) The energy U ( ) n depends on the nuclear coordinates and the electronic quantum number. The potential function V( x, ) complete potential function for fixed nuclear coordinates. Solve, change nuclear coordinates, solve again. Copyright Michael D. Fayer, 17
6 Solve electronic wave equation Nuclear Schrödinger equation becomes r j1 1 jn( ) n, Un( ) n ( ) M j Un( ) - the electronic energy as a function of nuclear coordinates,, acts as the potential function. Copyright Michael D. Fayer, 17
7 efore examining the hydrogen molecule ion and the hydrogen molecule need to discuss matrix diagonalization with non-orthogonal basis set. Two states and orthonormal No interaction With interaction of magnitude States have same energy: Degenerate The matrix elements are Copyright Michael D. Fayer, 17
8 amiltonian Matrix Matrix diagonalization - form secular determinant nergy igenvalues igenvectors Copyright Michael D. Fayer, 17
9 Matrix Formulation - Orthonormal asis Set eigenvalues N aij ij uj i 1, N j1 a u a u a u a u a u a u a u a u a u vector representative of eigenvector This represents a system of equations only has solution if a a a a a a a a a Copyright Michael D. Fayer, 17
10 asis Set Not Orthogonal but Normalized overlap asis vectors not orthogonal i j ij quals 1 if i = j. In Schrödinger representation * N aij ij uj i 1, N ij i j d system of equations j1 only has solution if (a (a a ) ) (a (a 1 a 3 1) 3 ) (a (a 13 3 a 13) 3) 33 Copyright Michael D. Fayer, 17
11 For a matrix with non-orthogonal basis set eigenvalues overlap integral, recover standard determinant for orthogonal basis. S 1 1 S 1 1 Copyright Michael D. Fayer, 17
12 ydrogen Molecule Ion - Ground State simple treatment r e r + r + orn-oppenheimer pproximation electronic Schrödinger equation m e e e 4r 4r 4r - refers to electron coordinates electron kinetic energy m ave multiplied through by Copyright Michael D. Fayer, 17
13 Large nuclear separations r System looks like atom and + ion nergy Rhc 13.6 ev Ground state wavefunctions U1s U1s ither atom at in 1s state with + at or atom at in 1s state with + at U1s and U degenerate 1s Copyright Michael D. Fayer, 17
14 Suggests simple treatment involving U1s as basis functions and U1s not orthogonal Form amiltonian matrix and corresponding secular determinant. eigenvalues overlap integral * U1s U 1s d * U1s U 1s d U U d * 1s 1s Copyright Michael D. Fayer, 17
15 nergies and igenfunctions S 1 1 S 1 U 1s 1s U 1 1s 1s U U S - symmetric (+ sign) - antisymmetric (- sign) (Not really symmetric and antisymmetric because only one electron.) Copyright Michael D. Fayer, 17
16 valuation of Matrix lements Need,, and * U1s U 1s d e e m 4 r 4r 4 r e Part of looks like ydrogen atom amiltonian e U1s U1s m 4r These terms operating on U1s can be set equal to U1s, - energy of 1s state of atom. Copyright Michael D. Fayer, 17
17 Then * e e U1s U1s d 4r 4r J e 4 ad * e J U1s U1s d 4r (Coulomb Integral) J e 1 1 a D D D e 1 4 D a r a h e distance in units of the ohr radius Copyright Michael D. Fayer, 17
18 * e e U1s U1s d 4r 4r (gain collecting terms equal to the atom amiltonian.) K e 4 a D * e K U1s U1s d 4r (xchange integral) e D K e 1 D 4 a (K is a negative number) D 1 e 1 D D 3 J - Coulomb integral - interaction of electron in 1s orbital around with a proton at. K - xchange integral exchange (resonance) of electron between the two nuclei. Copyright Michael D. Fayer, 17
19 These results yield S e J K 4 ad 1 e J K 4 ad 1 (K is a negative number) The essential difference between these is the sign of the exchange integral, K. lso consider e 1 e D N D 1 4a Classical - no exchange Interaction of hydrogen 1s electron charge distribution at with a proton (point charge) at. lectron fixed on. Copyright Michael D. Fayer, 17
20 classical no exchange -.8 repulsive at all distances anti-bonding M.O. -.9 e 8 a r -1. D e -1.1 N r S bonding M.O D r / a - equilibrium bond length S 1s energy e 1. 8 a e J K 4 ad 1 D e - dissociation energy e J K 4 ad 1 Copyright Michael D. Fayer, 17
21 Dissociation nergy quilibrium Distance D e r This Calc ev (36%) 1.3 Å (5%) xp..78 ev 1.6 Å Variation in Z'.5 ev (19%) 1.6 Å (%) Copyright Michael D. Fayer, 17
22 ydrogen Molecule r 1 e - e 1 - r r r 1 r 1 + r + In orn-oppenheimer pproximation the electronic Schrödinger equation is 1 m e e e e e e 4r1 4r1 4r 4r 4r1 4r Copyright Michael D. Fayer, 17
23 r, system goes to hydrogen atoms Use product wavefunction as basis functions U (1) U () I 1s 1s U () U (1) II 1s 1s lectron 1 around electron around. lectron around electron 1 around. These are degenerate. Copyright Michael D. Fayer, 17
24 Form amiltonian Matrix secular determinant I I II I I II II II II III II I II II d d * II I I 1 I I II II d d * III I II 1 I II II I * d d I II 1 Copyright Michael D. Fayer, 17
25 Diagonalization yields S II 1 III II 1 III S U U U U 1s 1s 1s 1s U U U U 1s 1s 1s 1s S - symmetric orbital wavefunction - antisymmetric orbital wavefunction Copyright Michael D. Fayer, 17
26 valuating attraction to nuclei * * e e IIU1s (1) U1 () s 4 r 4 r 1 nuclear-nuclear repulsion Two kinetic energy terms and two electron-nuclear attraction terms comprise atom amiltonians. e e 4r1 4r U (1) U () d d 1s 1s 1 electron-electron repulsion J J II J same as before. e 4 r e U1s (1) U1 () s J d 1d 4 r 1 e J e D D 4a D D D D r a / Copyright Michael D. Fayer, 17
27 * * e e IIIU1s (1) U1 () s 4 r 4 r 1 e e 4r1 4r U () U (1) d d 1s 1s 1 K K III e 4 r K and same as before. e U1s (1) U1 () 1 () 1 (1) s U s U s K d 1d 4 r 1 D r a / e D K e D 3D D a D ln D i 4D i D.577 (uler's constant) D 1 e 1 D D 3 i - integral logarithm math tables, approx., see book. Copyright Michael D. Fayer, 17
28 J and K - same physical meaning as before. J - Coulomb integral. ttraction of electron around one nucleus for the other nucleus. K - Corresponding exchange integral. J' - Coulomb integral. Interaction of electron in 1s orbital on nucleus with electron in 1s orbital on nucleus. K' - Corresponding exchange or resonance integral. Copyright Michael D. Fayer, 17
29 Putting the pieces together S 4 r 1 e J J K K 4 r 1 e J J K K K' is negative. The essential difference between these is the sign of the K' term. e N I I J J nergy of 4r U 1 U No exchange. Classical interaction of spherical atom charge distributions. Charge distribution about atoms and. lectron 1 stays on. lectron stays on. I 1s 1s Copyright Michael D. Fayer, 17
30 -1.6 classical no exchange repulsive at all distances anti-bonding M.O. oa o e N -.1 D e S -. r -.3 bonding M.O D r / a 1s energy e 1. 8 a This Calc. Dissociation nergy D e quilibrium Distance 3.14 ev (33%).8 Å (8%) xp. Variation in Z' 4.7 ev 3.76 ev (19%).74 Å.76 Å (3%) Vibrational Frequency - fit to parabola, 34 cm -1 ; exp, 4318 cm -1 r Copyright Michael D. Fayer, 17
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